Spectral concentration and rapidly decaying potentials

We consider the spectral function rho(mu) (<mu greater than or equal to 0) for the Sturm-Liouville equation y '' + (lambda - q)y = 0 on [0, infinity) with a boundary condition y(0) cos alpha + y'(0) sin alpha = 0 and where q is an element of L(0, infinity). A point mu(0) is called a point of spectral concentration if rho' has a local maximum there. We give a new formula for rho' in terms of the Prufer transformation angle theta and we identify a rapid transitional property of theta in the neighbourhood of points of spectral concentration. This property provides a sensitive test of the existence of such points and lends itself to a detailed computational investigation which reveals such points in greater profusion than shown by, for example, SLEDGE. Our approach also leads to the possibility of a theoretical investigation.